One of the most common construction in elementary geometry is drawing an altitude in a right angled triangle. This gives rise to 3 Symmetric triangles and one can use this to prove AM-GM inequality. However the same construction can also prove Pythagorean theorem. Here is this neat proof.
Sumant's 1 page of Math 
Pingback: Conway’s paper folding proof of Irrationality of Sqrt(2) | Sumant's 1 page of Math